Introduction to Decidability of Higher-Order Matching

  • Colin Stirling
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6014)


Higher-order unification is the problem given an equation t = u containing free variables is there a solution substitution θ such that t θ and u θ have the same normal form? The terms t and u are from the simply typed lambda calculus and the same normal form is with respect to βη-equivalence. Higher-order matching is the particular instance when the term u is closed; can t be pattern matched to u? Although higher-order unification is undecidable, higher-order matching was conjectured to be decidable by Huet [2]. Decidability was shown in [7] via a game-theoretic analysis of β-reduction when component terms are in η-long normal form.

In the talk we outline the proof of decidability. Besides the use of games to understand β-reduction, we also emphasize how tree automata can recognize terms of simply typed lambda calculus as developed in [1, 3-6].


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Colin Stirling
    • 1
  1. 1.School of InformaticsUniversity of Edinburgh 

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